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Is there an actual definition of "forgetful functor?" Wolfram MathWorld defines it in terms of functors from algebraic categories to the category of sets, but then says, "Other forgetful functors..." which seems an odd way to "define" something. In any event, it seems to exclude, say, the functor which takes topological groups to topological spaces (or groups.)

Wikipedia contains no definition, just lots of examples.

Is this one of those cases where we don't need a strict definition, we just recognize it when we see it?

Thomas Andrews
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    There is a little bit of discussion of your question here. – Cardboard Box Jun 30 '14 at 01:39
  • Personally I've always liked thinking of forgetful functors as inclusion functors, but this isn't entirely accurate (take for example the category of pointed topological spaces $\mathbf{Top}^\bullet$, which technically isn't a subcategory of $\mathbf{Top}$ but still has what is considered to be a forgetful functor to $\mathbf{Top}$). It's a naturally imprecise term, as Dane's link to nLab says. – Hayden Jun 30 '14 at 01:49
  • I think mostly the last paragraph, but try http://ncatlab.org/nlab/show/stuff%2C+structure%2C+property. – Qiaochu Yuan Jun 30 '14 at 01:55
  • @Hayden It also doesn't apply to the canonical "group to sets" forgetful function, since we can define many groups, potentially, on the same set... – Thomas Andrews Jun 30 '14 at 02:25
  • @ThomasAndrews That's very true, it seems my 'intuitive' definition covers less cases than I even thought. – Hayden Jun 30 '14 at 02:29
  • On the other hand, there is a natural isomorphism between the forgetful functor on groups and an inclusion functor, so that might be "good enough." @Hayden – Thomas Andrews Jun 30 '14 at 02:35
  • I'm voting to reopen because the 'duplicate' is asking a different question - it's much more specific and doesn't actually ask for a definition. This is a good question IMHO, and the answers are very helpful. – N. Virgo Feb 26 '20 at 09:13

2 Answers2

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The treatment on the nLab seems likely to be the state of the art here. There's no generally accepted definition, and to get one you might have to generalize to the stuff-structure-property perspective in which every functor can be seen as forgetful. This makes functors such as the projections $\mathcal C\times\mathcal D\to\mathcal C$ "forget stuff," namely, the $\mathcal D$ object of a pair.

If the idea of forgetting stuff feels unnatural, then maybe you want forgetful functors only to forget properties and structure-but this includes all faithful functors.

I think it's clear that every fully faithful functor $F:\mathcal D\to \mathcal C$ can reasonably be called forgetful, since it forgets the property of being in the subcategory $F(\mathcal D)$.

I don't see that there's such a natural description of what $F$ forgets when $F$ is merely faithful, but neither do I know of any way to specify when $\mathcal D$ is a category of $\mathcal C$-objects with structure, which might describe the most natural class of forgetful functors, better than the existence of a faithful functor $\mathcal D\to \mathcal C$. The comments under this question discuss the case of defining categories structured over sets, though not conclusively-but I'm at least convinced that there's no extant definition better than faithfulness.

So, it appears your options are "all functors," "faithful functors," or this.

Community
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Kevin Carlson
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    Yeah, I thought "faithful" was the best I was going to get, but the constant functor from $\mathcal C\to \mathcal C^2$ doesn't seem "forgetful," but it is faithful. – Thomas Andrews Jun 30 '14 at 02:47
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    @Thomas: you can make it seem forgetful if you play with it a little. Thinking of $C$ as the diagonal subcategory of $C^2$, the functor "forgets the property of being diagonal." (Like Kevin says, every fully faithful functor can be thought of as forgetting a property in this way.) – Qiaochu Yuan Jun 30 '14 at 06:06
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    One option not mentioned by Kevin is "faithful functor with a left adjoint." This is general enough to capture many important examples but not so general that it seems to lose all meaning. Unfortunately, it leaves a few examples out in the dust: for example, you'd probably be willing to agree with me that there's a forgetful functor from Hopf algebras to vector spaces, but as it turns out this functor doesn't have a left adjoint. – Qiaochu Yuan Jun 30 '14 at 06:09
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    An even more restrictive option which is a possible answer to Kevin's question about specifying when $D$ is a category of $C$-objects with structure is "monadic functor" (http://ncatlab.org/nlab/show/monadic+functor). But again, this definition is arguably too restrictive. For example, it fails to apply to the forgetful functor from topological spaces to sets. – Qiaochu Yuan Jun 30 '14 at 06:13
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I think that calling a functor forgetful is making a choice of a particular perspective on the source category, rather than declaring that the functor has some property.

When you choose to think of a functor $F : C \to D$ as forgetful, you're choosing to think of the objects in $C$ as objects in $D$ with extra structure in some vague sense. After all, from the perspective of abstract category theory $C$ is some opaque collection of objects and morphisms, which have no internal structure except what internal structure you choose to provide them, e.g. by choosing suitable forgetful functors.

(Incidentally, this is also an answer to a different question that one might want to ask about category theory, namely "what's the point of objects if all of the structure of a category is encoded in its morphisms?" One answer is that how you choose to describe the objects of a category can be thought of as declaring an intent to study a particular collection of forgetful functors out of your category. For example, choosing to describe the objects as sets with extra structure corresponds to declaring an intent to study a particular forgetful functor to $\text{Set}$.)

"Faithful functor" is not a terrible approximation but let me point out one example where it doesn't apply: many people would be happy to say that there is a forgetful functor from schemes to topological spaces, but this functor fails to be faithful. From my perspective, though, all that people mean when they say this is that they're declaring an intent to study schemes as topological spaces with extra structure (namely the structure sheaf).

Qiaochu Yuan
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    +1. I will never claim $\mathsf{Sch} \to \mathsf{Top}$ to be forgetful again. Actually this is one of the sources of confusion when one starts learning algebraic geometry, namely that (in contrast to manifolds) not just the space matters (also its sheaf!). Then one is inclined to think for example that the diagonal $\Delta(X) \subseteq X \times X$ equals ${z \in X \times X : \mathrm{pr}_1(z)=\mathrm{pr}_2(z)}$ (which is wrong). – Martin Brandenburg Jun 30 '14 at 07:56